Chapter 6 Applications of the Laplace Transform
Chapter 6 Applications of the Laplace Transform
Part One: Analysis of Network (6-2, 6-3)
Review of Resistive Network
1) Elements
[pic]
2) Superposition
[pic]
3) KVL and KCL
[pic]
4) Equivalent Circuits
[pic]
5) Nodal Analysis and Mesh Analysis
[pic]
Mesh analysis
[pic] Solve for I1 and I2.
Characteristics of Dynamic Network
Dynamic Elements ( Ohm’s Law: ineffective
1) Inductor
[pic]
2) Capacitor
[pic]
3) Example (Problem 5.9):
[pic]
Why so simple? Algebraic operation!
Dynamic Relationships (not Ohm’s Law) Complicate the analysis
Using Laplace Transform
[pic]
Define ‘Generalized Resistors’ (Impedances)
[pic] [pic]
[pic] As simple as resistive network!
Solution proposed for dynamic network:
All the dynamic elements ( Laplace Trans. Models.
( [pic] As Resistive Network
Key: Laplace transform models of (dynamic) elements.
Laplace transform models of circuit elements.
1) Capacitor
[pic]
Important: We can handle these two ‘resistive network elements’!
2) Inductor
[pic]
3) Resistor V(s) = RI(s)
4)Sources
[pic]
5) Mutual Inductance (Transformers)
[pic]
(make sure both i1 and i2 either away
or toward the polarity marks to make
the mutual inductance M positive.)
Circuit (not transformer) form:
[pic]
Benefits of transform
(
Let’s write the equations from this circuit form:
The Same
( Laplace transform model: Obtain it by using inductance model
[pic]
Just ‘sources’ and ‘generalized resistors’ (impedances)!
Circuit Analysis: Examples
Key: Remember very little, capable of doing a lot
How: follow your intuition, resistive network
‘Little’ to remember: models for inductor, capacitor and mutual inductance.
Example 6-4: Find Norton Equivalent circuit
[pic]
Assumption: [pic]
*Review of Resistive Network
1) short-circuit current through the load: [pic]
2) Equivalent Impedance or Resistance [pic]or [pic]:
A: Remove all sources
B: Replace [pic] by an external source
C: Calculate the current generated by the external source ‘point a’
D: Voltage / Current ( [pic]
*Solution
1) Find [pic]
[pic]
[pic]
2) Find [pic]
[pic]
condition: 1 ohm = 3/s
or I(s) = 0
=>I(s) = 0 =>Zs = (
3)
Example 6-5: Loop Analysis (including initial condition)
[pic]
Question: What are i0 and v0?
What is [pic]?
Solution
1) Laplace Transformed Circuit
[pic]
2) KVL Equations
[pic]
Important: Signs of the sources!
3) Simplified (Standard form)
[pic]
[pic]
4. Transfer Functions
1. Definition of a Transfer Function
1) Definition
[pic]
System analysis: How the system processes the input to form the output, or
[pic]
Input : variable used and to be adjusted
to change or influence the output.
Can you give some examples for input and output?
Quantitative Description of ‘ how the system processes
the input to form the output’: Transfer Function H(s)
[pic]
2) ( input
The resultant output y(t) to ( (t) input: unit impulse response
In this case: X(s) = L [( (t)] = 1
Y(s) = Laplace Transform of the unit impulse response
=> H(s) = Y(s)/X(s) = Y(s)
Therefore: What is the transfer function of a system?
Answer : It is the Laplace transform of the unit impulse response
of the system.
3) Facts on Transfer Functions
* Independent of input, a property of the system structure and parameters.
* Obtained with zero initial conditions.
(Can we obtain the complete response of a system based on its transfer
function and the input?)
* Rational Function of s (Linear, lumped, fixed)
* H(s): Transfer function
H( j2(f ) or H( j( ): frequency response function of the system
(Replace s in H(s) by j2(f or j()
|H( j2(f )| or |H( j( )|: amplitude response function
(H(j2(f) or (H( j( ): Phase response function
2. Properties of Transfer Function for Linear, Lumped stable systems
1) Rational Function of s
Lumped, fixed, linear system =>
[pic]
Corresponding differential equations:
[pic]
(2) [pic] all real! Why? Results from real system components.
Roots of N(s), D(s): real or complex conjugate pairs.
Poles of the transfer function: roots of D(s)
Zeros of the transfer function: roots of N(s)
Example: [pic]
(3) H(s) = N(s)/D(s) of bounded-input bounded-output (BIBO) stable
system
* Degree of N(s) ( Degree of D(s)
Why? If degree N(s) > Degree D(s)
[pic] where degree N’(s) X(s) = 1/s
[pic] ([pic] not bounded!)
* Poles: must lie in the left half of the s-plan (l. h. p)
i.e., [pic]
Why?
[pic]
(Can we also include k=1 into this form? Yes!)
[pic]
* Any restriction on zeros? No (for BIBO stable system)
3. Components of System Response
[pic]
Because x(t) is input, we can assume
[pic]
Laplace transform of the differentional equation
[pic]
D(s): System parameters
C(s): Determined by the initial conditions (initial states)
Initial-State Response (ISR) or Zero-Input Response (ZIR):
[pic]
Zero-State Response (ZSR) (due to input)
[pic]
From another point of view:
Transient Response: Approaches zero as t(∞
Forced Response: Steady-State response if the forced
response is a constant
How to find (1) zero-input response or initial-state response? No problem!
[pic]
(2) zero-state response? No prolbem!
[pic]
How to find (1) transient response? All terms which go to 0 as t((
(2) forced response? All terms other than transient terms.
Example 6-7
Input [pic]
Output [pic]
Initial capacitor voltage: [pic]
RC = 1 second
Solution
1) Find total response
[pic]
2) Find zero-input response and zero-state response
Zero-input response: [pic]
Zero-state response:
[pic]
3) Find transient and forced response
[pic]
Which terms go to zero as t((?
[pic]
What are the other terms:
[pic]
4. Asymptotic and Marginal Stability
System: (1) Asymptotically stable if [pic] as t(( (no input) for all
possible initial conditions, y(0), y’(0), … y(n-1)(0)
( Internal stability, has nothing to do with external input/output
(2) Marginally stable
[pic] all t>0 and all initial conditions
(3) Unstable
[pic]grows without bound for at least some values
of the initial condition.
(4) Asymptotically stable (internally stable)
=>must be BIBO stable. (external stability)
[pic]
6-5 Routh Array
1. Introduction
System H(s) = N(s)/D(s) asymptotically stable ( all poles in l.h.p (not
include jw axis.
How to determine the stability?
Factorize D(s):
[pic]
Other method to determine (just) stability without factorization?
Routh Array
1) Necessary condition
All [pic] (when [pic] is used)
⇨ any [pic] => system unstable!
Why?
Denote [pic] to esnure stability
[pic]
When all Re(pj) > 0 , all coefficients must be greater than zero. If some coefficient is not greater than zero, there must
be at least a Re(pj) system unstable
2) Routh Array
Question: All [pic] implies system stable?
Not necessary
Judge the stability: Use Routh Array (necessary and sufficient)
2. Routh Array Criterion
Find how many poles in the right half of the s-plane
1) Basic Method
[pic]
Formation of Routh Array
Number of sign changes in the first column of the array
=> number of poles in the r. h. p.
Example 6-8
[pic][pic]
sign: Changed once =>one pole in the r.h.p
verification:
[pic]
Example 6-9
[pic][pic]
[pic] [pic]
Sign: changed twice => two poles in r.h.p.
2) Modifications for zero entries in the array
Case 1: First element of a row is zero
⇨ replace 0 by ε (a small positive number)
Example 6-10
[pic][pic]
[pic]
Case 2: whole row is zero (must occur at odd power row)
construct an auxiliary polynomial and the perform differentiation
Example: best way.
Example 6-11
[pic][pic]
3) Application: Can not be replaced by MatLab
Range of some system parameters.
Example: [pic][pic]
[pic] [pic]
Stable system
[pic] to ensure system stable!
6. Frequency Response and Bode Plot
Transfer Function [pic]
Frequency Response
[pic]
Amplitude Response: [pic]
Real positive number: function of [pic]
Phase Response: [pic]
Interest of this section
In particular, obtain
[pic]
What are these?
[pic]
[pic]
Important Question: What is a Bode Plot?
How to obtain them without much computations?
Asymptotes only!
1. Bode plots of factors
1) Constant factor k:
[pic]
(2) s
[pic]
[pic]
Can we plot it?
[pic]: Can we plot [pic] for them?
Phase s:[pic]
[pic]: [pic]
[pic]
(3) [pic]
[pic]
step 1: Coordinate systems
step 2: corner frequency
[pic]
step 3: Label 0.1(c, (c , 10(c
step 4: left of (c : [pic]
step 5: right of (c : [pic]
Why?
[pic]
If
[pic]
If
[pic]
Example: [pic]
What is T : T = 0.2
What is (c : (c = 1/T = 5
Example : 0.2s + 1
Example : (0.2s + 1)2, (0.2s + 1)-2
Example : (Ts + 1)±N
(4) [pic] (Complex --- Conjugate poles)
Step 3 : Before [pic] : [pic]
Right of [pic] :
point 1: ([pic] , [pic])
point 2: ([pic] , [pic])
Example: [pic]
Actual [pic] and ( (show Fig 6-20)
What’s resonant frequency: reach maximum: [pic]
Under what condition we have a resonant frequency:
[pic]
[pic] : see fig 6-21
What about : [pic]?
2. Bode plots: More than one factors
[pic]
Can we sum two [pic] plots into one?
Can we sum two [pic] plots into one?
Yes!
3. MatLab
[pic]
Show result in fig 6-24
6.7 Block Diagrams
1. What is a block diagram?
Concepts: Block, block transfer function,
Interconnection, signal flow, direction
Summer
System input, system output
Simplification, system transfer function
2. Block
Assumption: Y(s) is determined
by input (X(s)) and block transfer
function (G(s)). Not affected by
the load.
Should be vary careful in
analysis of practical systems about the accuracy of this assumption.
3. Cascade connection
[pic][pic]
4. Summer
[pic]
5. Single-loop system
[pic]
[pic]
[pic]
[pic][pic]
Let’s find [pic][pic] Closed-loop transfer function
Equation (1) [pic][pic]
Equation (2) [pic][pic]
[pic][pic]
6. More Rules and Summary: Table 6-1
[pic]
Example 6-14: Find Y(s)/X(s)
[pic]
[pic]
[pic]
Example 6-15: Armature- Controlled dc servomotor
Input : Ea (armature voltage)
Output : [pic] (angular shift)
Can we obtain [pic]?
Example 6-16 Design of control system
[pic]
Design of K such that closed loop system stable.
[pic]
Routh Array: [pic][pic]
[pic] [pic]
System stable if k>0. If certain performance is required in addition to the stability, k must be further designed.
-----------------------
D(s)
[pic]
[pic]
C(s)
ZL
Why this direction?
Why this direction?
a
(Will I(s) be zero? We don’t know yet!)
Vtest(s)
[pic]
[pic]
N(s)
[pic]
[pic]
S
[pic]
[pic]
[pic]
Replace 0
Why?
[pic]
Line
Point 1
Point 2
[pic]
[pic]
[pic]
Line
[pic]
[pic]
[pic]
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- calculators in circuit analysis ed thelen
- ercot nodal protocols
- fundamentals of social network analysis
- nodal analysis lab michigan technological university
- notes 10 conductor sizing an example
- production engineering training hot hungary
- laboratory manual for dc electrical circuits
- nodal a system for ubiquitous collaboration
- chapter 6 applications of the laplace transform
Related searches
- applications of the necessary and proper clause
- 6 parts of the brain
- theoretical applications of the concept
- the outsiders chapter 6 questions
- the outsiders chapter 6 answers
- the outsiders chapter 6 pdf
- the outsiders chapter 6 summary
- the outsiders chapter 6 audio
- applications of the hierarchy of needs
- 6 steps of the scientific method
- 6 principles of the constitution
- ibc chapter 6 types of construction